Is this word onomatopoeic, just an interjection, both, something else?
It's not onomatopoeic in the normal sense, since onomatopoeia describes a word which mimics the thing it stands for. Unless hooray can satisfactorily be construed to mimic anything but itself and its own variants, it doesn't count as onomatopoeia. For instance, if it were possible to know that hooray did not descend from a word, and did descend from a natural and common noise, then it would be onomatopoeic.
On the surface, Wikipedia seems to favor the onomatopoeia hypothesis:
But even while it states that it is "probably onomatopoeic", it goes on to feed the other viewpoint, namely that hurrah did in fact originate from an arbitrary word, and therefore is not onomatopoeic.
The OED suggests ‘perhaps merely due to onomatopoeic modification, but possibly influenced by some foreign shouts’. It is given as a variant of ‘Hurrah’, itself coming in turn from ‘Huzza’, ‘apparently a mere exclamation, the first syllable being a preparation for, and a means of securing simultaneous utterance of the final /ɑː/ .
It isn’t onomatopoeic in quite the same way as ‘buzz’ or ‘hiss’ are, for the word is the same as what it describes, rather like ‘Hello’. It’s an interjection and there’s probably not much more to be said about it.
It's definitely not onomatopoeic. Onomatopoeia means words that "sound like" the thing being referenced - as, for example, neigh, oink, meow, or woof being used to represent sounds made by horses, pigs, cats, and dogs.
In my opinion, onomatopoeia is nowhere near as meaningful as many people think. The fact of the matter is those animal sound representations are almost entirely arbitrary, as evidenced by the fact that they're rarely even remotely similar across different languages.
If you didn't already know the conventions as to which orthographic representation goes with which animal, you'd probably never guess by just looking at the written form. Frankly, you'd have a hard time even if someone read them out to you unless they ridiculously exaggerated the pronunciation.
The only reason Hooray! (or Hurrah!) look like the sound people make when they're cheering is because we specifically replicate the written form we're already familiar with. People also shout Yay!, Hey! and Yeah! in the same context, or just holler some inarticulate sound.
As with a lot of interjections, such as Erm, Ho-hum, Doh, etc., these are just conventions that gradually come to be widely understood and replicated. Often there is no meaningful "origin".